Partial differential equations are paramount in mathematical modelling with applications in engineering and science. The book starts with a crash course on partial differential equations in order to familiarize the reader with fundamental properties…

This is a book about numerically solving partial differential equations occurring in technical and physical contexts and the authors have set themselves a more ambitious target than to just talk about the numerics. Their aim is to show the place of…

Large sparse linear systems of equations are ubiquitous in science, engineering and beyond. This open access monograph focuses on factorization algorithms for solving such systems. It presents classical techniques for complete factorizations that are…

The application of advanced analytics in science and technology is rapidly expanding, and developing optimization technics is critical to this expansion. Instead of relying on dated procedures, researchers can reap greater rewards by utilizing…

This open access book gives an overview of cutting-edge work on a new paradigm called the “sublinear computation paradigm,” which was proposed in the large multiyear academic research project “Foundations of Innovative Algorithms for Big Data.” That…

In this thesis we adapt fundamental parts of the Graph Minors series of Robertson and Seymour for the study of matching minors and investigate a connection to the study of directed graphs. We develope matching theoretic to established results of…

The problem of accurate and reliable simulation of turbulent flows is a central and intractable challenge that crosses disciplinary boundaries. As the needs for accuracy increase and the applications expand beyond flows where extensive data is…

The generalized Schur algorithm (GSA) allows computing well-known matrix decompositions,
such as QR and LU factorizations [1]. In particular, if the involved matrix is structured, i.e., Toeplitz, block-Toeplitz or Sylvester, the GSA computes the R…

This book, which presupposes familiarity only with the most elementary concepts of arithmetic (divisibility properties, greatest common divisor, etc.), is an expanded version of a series of lectures for graduate students on elementary number theory.…

First Semester in Numerical Analysis with Julia presents the theory and methods, together with the implementation of the algorithms using the Julia programming language (version 1.1.0). The book covers computer arithmetic, root-finding, numerical…

Integers are the building blocks of the theory of numbers. This chapter contains somewhat very simple and obvious observations starting with properties of integers and yet the proofs behind those observations are not as simple. In this chapter
we…

Elementary Differential Equations with Boundary Value Problems is written for students in science, engineering, and mathematics who have completed calculus through partial differentiation. If your syllabus includes Chapter 10 (Linear Systems of…

This book consists of ten weeks of material given as a course on ordinary differential equations (ODEs) for second year mathematics majors at the University of Bristol. It is the first course devoted solely to differential equations that these…

In order to apply mathematical methods to a physical or “real life” problem, we must formulate the problem in mathematical terms; that is, we must construct a mathematical model for the problem. Many
physical problems concern relationships between…

These notes were developed by the firsht author in the process of teaching a course on applied
numerical methods for Civil Engineering majors during 2002-2004 and was modified to include
Mechanical Engineering in 2005. The materials have been…